Relativity and Cosmology

Warp Drives are solutions of the Einstein Field Equations that allows superluminal travel within the framework of General Relativity. There are at the present moment two known solutions: The Alcubierre warp drive discovered in $1994$ and the Natario warp drive discovered in $2001$. However as stated by both Alcubierre and Natario themselves the warp drive violates all the known energy conditions because the stress energy momentum tensor(the right side of the Einstein Field Equations) for the Einstein tensor $G_{00}$ is negative implying in a negative energy density. While from a classical point of view the negative energy is forbidden the Quantum Field Theory allows the existence of very small amounts of it being the Casimir effect a good example as stated by Alcubierre himself.The major drawback concerning negative energies for the warp drive are the so-called Quantum Inequalities(QI) that restricts the time we can observe the negative energy density.This time is known as the sampling time.Ford and Pfenning computed the QI for the Alcubierre warp drive using a Planck Length Scale shape function and concluded that the negative energy in the Alcubierre warp drive can only exists for a sampling time of approximately $10^{-33}$ seconds rendering the warp drive impossible for an interstellar trip for example a given star at $20$ light years away with a speed of $200$ times faster than light because such a trip would require months not $10^{-33}$ seconds. We repeated the QI analysis of Ford and Pfenning for the Natario warp drive using the same Planck Length Scale but with a shape function that although different from the function choosed by Ford and Pfenning it obeys Natario requirements and because the Natario warp drive have a very different distribution of negative energy when compared to its Alcubierre counterpart this affects the QI analysis.We arrived at a sampling time that can last longer than $10^{-33}$ seconds enough to sustain a warp bubble for the interstellar travel mentioned above.We also computed the total negative energy requirements for the Natario warp drive and we arrived at a comfortable result.This leads us to conclude that the Natario warp drive is a valid solution of the Einstein Field Equations of General Relativity physically accessible for interstellar spaceflight. We also discuss Horizons and infinite Doppler blueshifts.

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